Solution of matrix games with payoffs of single-valued trapezoidal neutrosophic numbers

Single-valued neutrosophic numbers (SVNNs) are very much useful to express uncertain environments. In real-life problems, there are many situations where players of a matrix game can not assess their payoffs by using ordinary fuzzy sets or intuitionistic fuzzy sets. In these situations, single-valued trapezoidal neutrosophic numbers (SVTNNs) play a vital role in game theory, as it includes indeterminacy in the information besides truth and falsity. The objectives of this paper are to explore matrix games with SVTNN payoffs and to investigate two different solution methodologies. To solve such games, a pair of neutrosophic mathematical programming problems have been formulated. In the first approach, the two neutrosophic mathematical programming models are converted into interval-valued multi-objective programming problems by using a new ranking order relation of SVTNNs. Finally, the reduced problems are solved using the weighted average approach and utilizing LINGO 17.0 software. It is worth mentioning that the values of the game for both the players are obtained in SVTNN forms, which is desirable. In the second approach, each neutrosophic mathematical programming model is transformed into a crisp one by using the idea of \(\alpha \)-weighted possibility mean value for SVTNNs. A market share problem and another numerical example are illustrated to show the validity and applicability of the proposed approaches.